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An Asymptotic (4/3+ε)-Approximation for the 2-Dimensional Vector Bin Packing Problem [article]

Ariel Kulik, Matthias Mnich, Hadas Shachnai
2022 arXiv   pre-print
We give an asymptotic (4/3+ε)-approximation for the problem, thus improving upon the best known asymptotic ratio of (1+ln3/2+ε)≈ 1.406 due to Bansal, Elias and Khan (SODA 2016).  ...  We study the 2-Dimensional Vector Bin Packing Problem (2VBP), a generalization of classic Bin Packing that is widely applicable in resource allocation and scheduling.  ...  Our main result is an improved asymptotic approximation ratio for the problem. Theorem 1.1. For any ε > 0 there is a polynomial-time random asymptotic 4 3 + ε -approximation for 2VBP.  ... 
arXiv:2205.12828v1 fatcat:bo4rqxmmk5fvpk3xz5kyk4mu7a

Tight Approximation Algorithms for Two Dimensional Guillotine Strip Packing [article]

Arindam Khan, Aditya Lonkar, Arnab Maiti, Amatya Sharma, Andreas Wiese
2022 arXiv   pre-print
Moreover, due to a reduction from the Partition problem, it is NP-hard to obtain a polynomial-time (3/2-ε)-approximation algorithm for GSP for any ε>0 (exactly as Strip Packing).  ...  This problem generalizes the classical Bin Packing problem and also makespan minimization on identical machines, and thus it is already strongly NP-hard.  ...  There is a pseudo-polynomial time (4/3 + ε)-approximation [22] for 2GK.  ... 
arXiv:2202.05989v3 fatcat:bzrmvh3gmbbjlpw6jdd5xn3p4y

Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items [article]

Arindam Khan, Eklavya Sharma
2021 arXiv   pre-print
The conjecture, if true, will imply a (4/3+ε)-approximation algorithm for 2BP.  ...  In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into  ...  An Asymptotic Polynomial-Time Approximation Scheme (APTAS) is an algorithm that accepts a parameter ε and has AAR of (1 + ε). 2BP is a generalization of classical 1-D bin packing problem [22, 15] .  ... 
arXiv:2105.02827v1 fatcat:jzjpllwxhzdffjiao7go3epemq

Approximation Algorithms for Rectangle Packing Problems (PhD Thesis) [article]

Salvatore Ingala
2017 arXiv   pre-print
Thus, we consider a generalized kind of packing that combines container packings with another packing problem that we call L-packing problem, where we have to pack rectangles in an L-shaped region of the  ...  In 2-Dimensional Geometric Knapsack (2DGK), the target region is a square of a given size, and our goal is to select and pack a subset of the given rectangles of maximum weight.  ...  Chapter 3 A PPT (4/3 + ε)-approximation for Strip Packing In this chapter, we present a ( 4 3 + ε)-approximation for Strip Packing running in pseudo-polynomial time.  ... 
arXiv:1711.07851v1 fatcat:awwgt5x74fbfzim5mmsmndrlhu

Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More [article]

Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, Andreas Wiese
2021 arXiv   pre-print
In this paper, we give (4/3+ε)-approximation algorithms in polynomial time for both cases, assuming that all input data are integers polynomially bounded in n. Gálvez et al.'  ...  In the 2-Dimensional Knapsack problem (2DK) we are given a square knapsack and a collection of n rectangular items with integer sizes and profits.  ...  , so that there exists a feasible packing of a (4/3 + ε)-approximate solution into them.  ... 
arXiv:2103.10406v1 fatcat:gn6xqsyugjc37ic53xret5xxlm

A PTAS for Packing Hypercubes into a Knapsack [article]

Klaus Jansen, Arindam Khan, Marvin Lira, K. V. N. Sreenivas
2022 arXiv   pre-print
For this problem, Harren (ICALP'06) gave an algorithm with an approximation ratio of (1+1/2^d+epsilon).  ...  As a side result, we give an almost optimal algorithm for a variant of the strip packing problem in higher dimensions.  ...  Acknowledgements We thank Roberto Solis-Oba and three anonymous reviewers for their comments and suggestions on this paper.  ... 
arXiv:2202.11902v2 fatcat:ekizmcgxvbg6dgjw7ma7cmrwnu

Approximating Geometric Knapsack via L-packings [article]

Waldo Gálvez and Fabrizio Grandoni and Sandy Heydrich and Salvatore Ingala and Arindam Khan and Andreas Wiese
2017 arXiv   pre-print
We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack.  ...  The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004].  ...  Open Problems The main problem that we left open is to find a PTAS, if any, for 2DK and 2DKR. This would be interesting even in the cardinality case.  ... 
arXiv:1711.07710v1 fatcat:wyl6uu4dercwxldztvrphpfdki

Closing the gap for pseudo-polynomial strip packing [article]

Klaus Jansen, Malin Rau
2019 arXiv   pre-print
The set of 2-dimensional packing problems builds an important class of optimization problems and Strip Packing together with 2-dimensional Bin Packing and 2-dimensional Knapsack is one of the most famous  ...  The strength of this structural result is that it applies to other problem settings as well for example to Strip Packing with rotations (90 degrees) and Contiguous Moldable Task Scheduling.  ...  On the other hand, the algorithm with the best ratio so far computes a 4/3 + ε approximation [8, 16] .  ... 
arXiv:1712.04922v2 fatcat:4dsd556ud5cafi6n6r2wh367vq

Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More

Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, Andreas Wiese, Kevin Buchin, Éric Colin de Verdière
2021
In this paper, we give (4/3+ε)-approximation algorithms in polynomial time for both cases, assuming that all input data are integers polynomially bounded in n. Gálvez et al.'  ...  In the 2-Dimensional Knapsack problem (2DK) we are given a square knapsack and a collection of n rectangular items with integer sizes and profits.  ...  so that there exists a feasible packing of a (4/3 + ε)-approximate solution into them.  ... 
doi:10.4230/lipics.socg.2021.39 fatcat:u2za5n3xfzbnjf7z7zczzoe7iy

Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items

Arindam Khan, Eklavya Sharma, Mary Wootters, Laura Sanità
2021
The conjecture, if true, will imply a (4/3+ε)-approximation algorithm for 2BP.  ...  In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into  ...  Vector packing (VP) is another variant of bin packing, where the input is a set of vectors in [0, 1] d and the goal is to partition the vectors into the minimum number of parts (bins) such that in each  ... 
doi:10.4230/lipics.approx/random.2021.22 fatcat:akjmxzmscrcftfml5xzu3o74ei

A PTAS for Packing Hypercubes into a Knapsack

Klaus Jansen, Arindam Khan, Marvin Lira, K. V. N. Sreenivas, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruff
2022
For this problem, Harren (ICALP'06) gave an algorithm with an approximation ratio of (1+1/2^d+ε).  ...  As a side result, we give an almost optimal algorithm for a variant of the strip packing problem in higher dimensions.  ...  Acknowledgements We thank Roberto Solis-Oba and three anonymous reviewers for their comments and suggestions on this paper.  ... 
doi:10.4230/lipics.icalp.2022.78 fatcat:yon2tzq3vnfhla44l52w7ncaqq